AC loss in pancake coil made from 12 mm wide REBCO tape

IEEE/CSC & ESAS EUROPEAN SUPERCONDUCTIVITY NEWS FORUM, No. 22, October/November 2012. 

AC loss in pancake coil made from 12 mm wide 

REBCO tape 

Fedor Gömöry, Ján Šouc, Enric Pardo, Eugen Seiler, Mykola Soloviov, Lubomír Frolek, Michal Skarba, 

Pavol Konopka, Marcela Pekarčíková, Jozef Janovec 

Abstract— The design of a superconducting coil from high 

performance REBCO coated conductors is often complicated 

because of the complex anisotropy of the critical current density, 

Jc. It is important to understand how much in detail this feature 

must be taken into consideration in the prediction of maximum 

achievable current and the expected AC loss. We present the 

results of investigation performed with a small (10 turns, 60 mm 

inner diameter) coil made from SuperPower tape of 12 mm 

width. The knowledge of Jc(B,�) determined on short sample 

allowed to predict the maximum achievable current of the coil 

and the AC loss behavior with surprising accuracy. We have also 

investigated the effect of a tape non-uniformity. Our results 

confirm that the lateral non-uniformity when J c at tape edges is 

lower than in its centre leads to significant increase of AC loss. A 

longitudinal non-uniformity, in particular the reduction of 

critical current in some portion along the tape length is hardly 

observable in AC loss result. Regarding the achievable coil 

current only the innermost turn is critical, thus using a piece of 

tape with lower J c there would significantly reduce the maximum 

current. We present also calculations showing the change in 

current-voltage curve and redistribution of AC dissipation in the 

case of non-uniform tape quality. 

� 

Index Terms—superconducting magnets, high-temperature 

superconductors, AC losses, numerical computations 

C 

I. INTRODUCTION 

OATED conductor (CC) tapes from high temperature 

superconductors (HTS) are more and more available in 

lengths and amperages that allow considering a replacement of 

copper electromagnets in different kinds of electric power 

devices (transformers, motors) by coils from HTS CC. There 

are several significant differences between these two materials 

(working temperature, shape and sensitivity to mechanical 

strain, …). From the electromagnetic point of view the HTS 

material is completely dissimilar to a normal metal that is well 

characterized by its electrical resistivity. The conductivity of 

HTS depends in highly non-linear way on the electric and 

magnetic field and thence the distribution of current is 

Manuscript received October 9, 2012. This work was supported in part by 

the VEGA grant agency under contract N. 1/0162/11 and by the European 

Commission EURATOM project FU07-CT-2007-00051. 

F. Gömöry, J. Šouc, E. Pardo, E.Seiler, M. Soloviov, L. Frolek are with the 

Institute of Electrical Engineering, Slovak Academy of Sciences, Dubravska 

cesta 9, 84104 Bratislava, Slovakia (corresponding author phone: +4212 5922 

2033; fax: +4212 5477 5816; e-mail: elekgomo@savba.sk). M. Skarba, P. 

Konopka, M. Pekarčíková, J. Janovec are with Faculty of Materials Science 

and Technology in Trnava, Slovak Technical University, Paulínska 16, 917 24 

Trnava, Slovakia. 

1 of 7 

Fig. 1. Photograph of the coil (top) assembled with current leads and 

voltage taps and (bottom) its geometry used in calculations: 

Rin = 60 mm, dR = 0.4 mm, wSC = 12 mm. Number of turns is 10. 

generally non-uniform. As a result the procedure of designing 

a coil from HTS as well as the interpretation of results from 

its experimental testing requires particular rules to be obeyed. 

We illustrate this approach on the example of a small coil (10 

turns, inner diameter 60 mm) made from 12 mm wide CC tape 

[1]. Individual turns were insulated by textile tape 0.25 mm 

thick and then the winding was impregnated by polyesterimide 

resin dissolved in styrene [2]. The winding was self-supported 

– Fig. 1 - but to some extent deformable without damage. We 

assume that CC tapes did not experience significant 

mechanical load during thermal cycling. 

Coated conductor coils have been the object of study of 

several works, either by measurements [3, 4, 5, 6, 7] or 

computations [4, 6, 7, 8, 9, 10]. However, the effect of 

degradation of the tape across its width or its length has not 

been studied. This article not only presents such study but also 

a full systematic for the characterization and numerical 

simulation of pancake coils. 

In the first part of the paper we show how the coil 

performance can be predicted from the data obtained on a


Fig. 2 Short-sample data (symbols) of the critical current dependence on the magnetic field magnitude (different symbol for each value explained in the legend) 

and its orientation with respect to the CC tape wide face, defined by the angle � . Empirical fits with the expression (1) are shown by lines. The measured 

dependence has explicit 360� periodicity at low magnetic fields, however beyond 100 mT the values for 90�-270� could be considered a repetition of those 

obtained in the -90� - 90� range. 

short sample. Because of highly non-linear properties of HTS 

tape it is necessary to employ numerical computation 

technique in the design of the coil. It is also useful in analysis 

of the observed behavior and allows to calculate the quantities 

that are not easily accessible in experimental way like the 

distribution of heat generated by AC current in the coil. 

II. TAPE PROPERTIES 

Basic information in the design of a superconducting coil is 

the dependence of critical current on magnetic field and its 

orientation. In the case of SuperPower tape SCS12050-AP this 

dependence is rather complicated as shows Fig. 2. Explanation 

of this complex dependence is beyond the scope of our 

investigation. However its description is essential in the coil 

design and performance prediction. We have used several 

approximations [11], the results presented here have been 

obtained by using the empirical description: 

2 

� �� 

�� 

� � 

0( 

B) 

� I 2( 

B) 

�1 

� � � � 

� �1� 

� 

� � �� 

� � I1( 

B) 

� 

I1( 

B) 

� 

J c ( B, 

� ) � � � 

for � �� 

0 � � 

SSC 

� 

� 

�� 

1 for � �� 

0 � �� 

where �� = 0.75 radians and 

(1) 

� 

2 of 7 

I ( B) 

� I 

1 

I ( B) 

� I 

2 

min 0 

max 0 

� 

�1� 

k 

�� 

� 

�1� 

k 

�� 

min 

max 

� � 

� � 

� 

� 

0 ( B) 

� �00 

1� 

k� 

1� 

�� 

� 

� 

� 

�1� 

� 

� 

� 

�1� 

� 

B 

B 

0 

� 

� 

� 

� 

with the parameters B0 = 0.5 T, � = 4.5, Imin0 = 136 A, Imax0 = 

218 A, �00 = 0.58 radians and SSC is the cross-section of the 

Critical current [A] 

400 

380 

360 

340 

320 

300 

280 

260 

240 

220 

200 

B 

B 

0 

B 

B 

� 

� 

� 

� 

� 

0 

� 

� 

�� 

� 

� 

� 

� 

� 

1 2 3 4 5 6 7 8 9 10 

Turn number 

Fig. 3. Critical currents of coil turns estimated with the help of (1). Turns are 

numbered from inside towards outside, the innermost turn has number 1. 

� 

� 

�� 

� 

� 

� 

��


0.001 

E coil 

[V 

] 

/m 

0.0001 

Experiment 

Simulation 

Simulation - 2% degradation all turns 

Simulation - 7% degradation turn 1 

Simulation - 7% degradation all turns 

0.00001 

200 Icoil [A] 

300 

Fig. 4. Current-voltage dependence calculated for the coil (full line) 

compared with the experimental data (circles). Better agreement is reached 

assuming the critical current data lowered by 2% (dotted). Simulations for a 

7% degradation of the innermost turn (long dashes) and of all the coil turns 

(short dashes) were carried out to see how much such damage could be 

detected in the E(I) tests. 

superconducting layer that in our calculations was represented 

by the rectangle of width wSC = 12 mm and thickness hSC = 

1 �m. The expression (1) with these parameters fits the data 

for angles � from -90� to 90�and magnetic fields higher than 

0.1 T, the critical currents for lower fields are slightly higher 

in the range of angles � between 90� and 270�. 

III. MAXIMUM DC CURRENT OF THE COIL 

Knowledge of the tape critical current described by the 

expression (1) allows to predict the maximal DC current the 

coil could carry. For this purpose we created the finite-element 

model based on the geometry shown in Fig. 1. Rotational 

symmetry is used i.e. the real spiral turns have been replaced 

by circular ones. The approximation of uniform current 

density in the winding was used to calculate the magnetic field 

distribution in the coil. Then in an iterative procedure the level 

of coil current that would generate the electrical field Ec = 

1 μV/cm in various turns has been identified. The result – that 

could be with certain simplification called the critical currents 

of individual turns – is shown in Fig. 3. As expected, the 

innermost turn (assigned No. 1) limits the maximum DC 

current the coil could carry. Then it does not make much sense 

to calculate the current that would create 1 μV/cm electrical 

field in the whole coil. Instead we estimated the currentvoltage 

curve of the whole coil. A rigorous calculation would 

require in the finite element calculation a dense mesh of 

current-voltage dependences taken on short sample at different 

field values and orientations. We used a simplified procedure 

instead: First, the measured short sample curves were 

approximated by the power-law function � �n E ( I ) � E c I I c 

(from which the values of Ic found at Ec = 1 μV/cm were used 

in Fig. 2) where the exponent n was found to vary between 17 

and 25 in the range of magnetic fields 0.08-0.14 T that would 

appear in the coil winding at carrying maximum current. Then 

we took the “critical currents” of individual turns, Ici, and the 

power-law exponent in the mid of its range, n = 21 to estimate 

the current-voltage curve of the coil: 

E 

coil 

( I 


) 

N 

� 

� 

i�1 

i 

N 

� I 

2�r 

E 

� 

� c 

� I 

� 

i�1 

2�r 

i 

In this expression N = 10 is the number of turns, each of them 

at the radius ri. The result was confronted with experimental 

data as shown in Fig. 4. Calculated prediction is slightly 

higher than the experimental finding. Key role of the 

innermost turn is illustrated by E(I) curve calculated 

considering the degradation of critical current by e.g. 7%: The 

shift towards lower currents obtained for a uniform reduction 

of Ici in all the turns is close to the result obtained assuming 

such degradation in the innermost turn alone. 

IV. AC LOSS BEHAVIOR 

In the case of a coil considered for power application the 

dissipation at AC transport (AC loss) is an important 

parameter. We performed numerical calculations based on the 

A-�-j formulation of critical state utilizing the finite-element 

code Comsol Multiphysics [12]. The original stepwise currentvoltage 

curve proposed for hard superconductor [13] is 

replaced by its smooth approximation 

J 

3 of 7 

s, 

� 

( r, 

z) 

� J 

c 


ci 

� 

� 

� 

� 

n 

(2) 

� E� 

( r, 

z) 

� 

tanh 

� 

� 

� 

� . (3) 

� Ec 

� 

where the critical current density Jc is now the upper limit for 

the current density in superconductor, Js, and the cylindrical 

geometry shown in Fig.1 was introduced. Because the 

electrical field is generated by the electrostatic gradient � � 

as well as the time variation of the magnetic vector potential, 

A � , the expression (3) can be rewritten [6] as 

� 

�A� 

( r, 

z) 

� 

� � �� 

( r, 

z) 

� � 

� 

� 

�t 

� (4) 

J s, 

� ( r, 

z) 

J c ( B, 

� ) tanh 

� Ec 

� 

� 

� 

� 

� 

. 

where we also indicated that Jc depends on the local value of 

magnetic field and its orientation with respect to the tape. In 

the finite element calculation the distribution of A and �� is 

found that obeys the Maxwell equations with the additional 

constraint that the total current in each turn is equal to the 

current delivered by the power supply

� 

S 


i 

J 

s, 

� 

( r, 

z) 

i �1.. 

N 

� I 


Series of distributions are calculated for 

I � I �t ) � I sin( � ) 

(6) 

coil , j max sin( j max j 

where � is the angle frequency and �j = 0, �.�tj, 2�.�tj,,… 2� 

are the angles corresponding to the time instants tj when the 

distributions are resolved. We have found that calculation of 

M = 40 distributions in one AC period is sufficient to obtain 

the representative result within reasonable computation time. 

The power dissipated in the coil at the time instant tj is the 

sum of contributions from individual turns 

P 

j 

� N 

� � 

i�1 S 

i 

J ( r, 

z) 

E ( r, 

z) 

dS 

(7) 

s 

i, 

j 

where Js is calculated using (4) and the electrical field in the ith 

turn at the instant defined by �j is 

E 

Q [J/m] 

i, 

j 

1.E‐01 

1.E‐02 

1.E‐03 

1.E‐04 

1.E‐05 

10 100 Imax [A] 

100 

( r, 

z) 

� �� 

i, 

j 

A ( r, 

z) 

� A ( r, 

z) 

� � 

� � � 

� j � j �1 

(8) 

j 

j �1 

Experiment 

Jc(B) 

Jc(B) + Nonuniformity 

Fig. 5. AC loss in the coil calculated by finite-element method (lines) 

compared with experimental data (symbols). Taking into account the Jc(B,�) 

is not enough to reproduce the measurements (dashed line). Sufficient 

agreement was obtained when the lateral non-uniformity in the critical 

current density was considered (full line). 

In the last expression we took into account that �� is constant 

in the cross-section of one turn and it thus attains discrete 

values��i for different turns [14, 15]. The power dissipated in 

all the turns is integrated over one cycle to obtain the total loss 

in the winding. In order to allow comparison with results 

obtained on different coils we prefer to present the dissipation 

(5) 

4 of 7 

J c [A/m 2 ] 

divided by the total length of the tape in Joule/meter units: 

Q 

2.3 

2.1 

1.9 

1.7 

1.5 

1.3 

1.1 

0.9 

0.7 

x 1e10 

0.5 

‐6 ‐4 ‐2 0 2 4 6 

position with respect to the tape centre [mm] 

Fig. 6. Nonuniformity of the tape across its width illustrated by (top) profile 

of the critical current density and (bottom) morphology of the 

superconducting layer. The profile of Jc (full line) was obtained from the 

measured distribution of magnetic field above the tape carrying critical 

current at DC background field of 250 mT. Its linear approximation is shown 

by the dashed line. Electron microscope pictures were taken after remowing 

the protecting Ag and stabilizing Cu layer. The inhomogenities – seen as 

“islands” in the superconducting layer - occupy bigger portion at tape edges. 

2� 

M 

M 

� Pj 

�t 

� Pj 

j �1 

�M 

j �1 

� � N 

Lcoil 

� 2�ri 

i �1 

The prediction obtained by such calculation incorporating 

the Jc(B,�) expression (1) is compared in Fig. 5 with 

experimental data determined by standard lock-in technique. 

The loss per cycle measured at frequencies of 36, 72 and 144 

Hz, respectively, was identical testifying pure hysteretic 

character consistent with the critical state model. As one can 

see, the calculated prediction (dashed line) was below the 

experimental data. The observed discrepancy was removed 

when we took into account the fact that the tape edges exhibit 

(9)


Q [J/m] 

1.E‐01 

1.E‐02 

1.E‐03 

1.E‐04 

Experiment 

Calculation, Ic = 260 A 



1.E‐05 

10 100 Imax [A] 

1000 

Fig. 7. AC loss in the coil calculated in simplified manner assuming a 

constant critical current density, Jc, (lines) compared with experimental data 

(symbols). The results are shown for three values of Jc corresponding to the 

tape critical currents shown in the legend. 

reduced current transport capability compared to the central 

part. Such reduction was observed in separate experiment 

mapping the magnetic field distribution above the tape 

transporting DC current. The variation of Jc across the tape 

width shown in Fig. 6 has been derived by an inversion 

calculation [16]. Also the structural investigations we 

performed by scanning electron microscopy revealed that 

there is significant difference in the superconducting layer 

morphology across the tape width. The structural analysis 

suggests that the volume ratio of inhomogeneities in the 

superconducting layer is increased towards both edges of the 

tape. Such non-uniformity can never be deduced from the 

Ic(B,�) data. On the other hand at AC transport the dissipation 

depends on details of the flux front movement resulting in 

significant difference between two calculated curves in Fig. 5. 

However, the coincidence of the measured AC loss 

data with the calculation is not always the proof of a valid 

description of superconductor properties. We performed a 

series of AC loss calculations neglecting the magnetic-field 

dependence of the critical current density. Main motivation for 

such exercise was to check how much the improvement (or 

degradation) of the critical current would affect the AC loss. 

Here we show the results for three values of Jc that for the 

considered cross-section of superconducting layer (12 mm 

wide and 1 µm thick) correspond to the critical currents equal 

to 260, 334 and 390 A, respectively. The lowest and highest of 

these values are the extreme levels of critical currents that 

were calculated for individual turns in the coil taking the 

complex Ic(B,�) of the tape into account. The middle value is 

obtained from an average of these values shown in Fig. 2. 

The main conclusion is that the coil loss calculated 

assuming a field-independent critical current density Jc = 

const. – see Fig. 7 - is inversely proportional to Jc. Such 

behavior is typical for the self-field (transport) regime [17]. 

Maybe in a coil with significantly larger number of turns this 

5 of 7 

conclusion could not apply. Rather surprising however is that 

the result calculated in such simplified manner - assuming that 

the critical current density in all the winding is equal to that of 

the innermost turn - does very well match the experimental 

data. We believe this is a mere coincidence: As shown by the 

dashed line in Fig. 5 the slope of the loss curve obtained 

assuming Jc(B) dependence - without lateral non-uniformity - 

would be higher than that one resulting from a Jc = const. 

assumption. 

As we have found, the lateral non-uniformity of Jc 

with respect to the tape width has significant influence on the 

coil loss. This effect has already been found in the case of 

power transmission cable [18] or fault current limiter [19]. In 

the following we present the results of our analysis about the 

possible consequence of Jc variation along the tape length. In 

particular we calculated the AC loss and its distribution in the 

winding in the case when Jc in one of the turns is lower than in 

the rest. 

V. INFLUENCE OF LOCAL I C DEGRADATION 

In this section we show the results of AC loss calculation 

performed following the procedure described by the formulas 

(4) to (9) i.e. taking into account the complex Jc(B,�) 

established on the short sample assuming however that the 

superconductor in some turns has Jc that is proportionally 

lowered at all the field magnitudes and orientations. 

First notable feature is that the AC loss is far less sensitive to a 

single turn degradation than the current-voltage curve. This is 

obvious when comparing the effect of reducing the critical 

current of the innermost turn to 93% of original value on the 

current-voltage curve in Fig. 4 with the AC loss result shown 

in Fig. 8. Even a drop to 75% of the original value does not 

produce an increase of AC loss in the coil that would be 

distinctively beyond the experimental noise. 

1.E‐04 

Q [J/m] 

1.E‐05 

10 

No degradation 

7% degradation of turn 1 

25% degradation of turn 1 

I max [A] 

Fig. 8. Calculated effect of a reduced critical current in the innermost turn on 

AC loss: loss in the coil (full line from Fig. 5) compared with the result 

obtained assuming critical current of the turn N.1 reduced to 93% and 75%, 

respectively. As one can see, such degradation produces an effect that is 

hardly visible. 

50


Loss per cycle [J] 

2.E‐04 

1.E‐04 

0.E+00 

1 2 3 4 5 6 7 8 9 10 

Turn number, i 

This result can be better understood when we analyze 

the distribution of dissipation in the coil winding. In Fig. 9 the 

dissipation is plotted as calculated for individual turns in the 

AC cycle with amplitude of 70 A. The dissipation is rather 

uniformly distributed inside the coil, the extreme turns (the 

innermost and the outermost ones) producing heat about 50% 

above the average. Then one can better understand why a 

reduction of the critical current in one turn causes a minor 

increase of the dissipation in the whole coil. It is interesting is 

that an increase of dissipation is predicted also for the 

neighboring turns in spite of the fact that their properties have 

not changed. Surprising is the result we have found for higher 

AC amplitudes as shown in Fig. 10: In the case of coil made 

of tape with the same properties in all the turns the dissipation 

pattern is similar to that found at lower currents. Striking is the 

prediction that using a degraded tape in the innermost turn 

would reduce the dissipation in that turn at higher currents. 

The elevated AC loss in the whole coil should be borne by 

nearly all the remaining turns, the additional dissipation 

peaking at the neighboring turn N.2. We have checked this 

prediction also by a different numerical method based on the 

critical state principles [10] with identical results. 

Unfortunately there is no easy way to detect the local 

dissipation in coil turns in experiment. 

VI. CONCLUSION 

Undegraded Ic 

75% Ic in Turn 1 

Fig. 9. Distribution of dissipation in the coil turns at AC amplitude of 70 A 

(27% of undegraded Ic of inner turn): Line with diamonds shows the values 

calculated for the coil made of identical turns from the tape with properties 

shown in Fig.2. The full squares have been obtained assuming that in the 

innermost turn the critical current has been lowered by 25%. 

Properties of a small coil made from 12 mm wide CC tape 

have been investigated experimentally and by finite element 

numerical models. Taking into account the complexity of Jc 

dependence on magnetic field and its orientation in the 

calculation we achieved rather precise prediction of the DC 

current-voltage curve of the coil as well as the dissipation in 

AC regime. 

6 of 7 

Loss per cycle [J] 

1.5E‐03 

1.0E‐03 

5.0E‐04 

Undegraded Ic 

75% Ic in Turn 1 

1 2 3 4 5 6 7 8 9 10 

Turn number, i 

Fig. 10. The same as Fig. 9 but the AC amplitude is now 140 A (54% of 

undegraded Ic of inner turn). Surprising is that the turn with lowered critical 

current exhibits lower dissipation than in the case it were undegraded. The 

increased dissipation is distributed in the remaining coil turns. 

Similar to the results of other groups analyzing the AC loss 

in superconducting cables or fault current limiters we have 

found that a non-uniform transport capability of the HTS 

material across the tape width leads to significant increase in 

AC dissipation. Our structural investigations provide strong 

indication that this phenomenon is caused by increased share 

of “islands” not participating in the current flow at the tape 

edges. 

Longitudinal non-uniformity of tape critical current can 

have severe consequences on the maximum current it is able 

to reach. On the other hand the AC loss is rather insensitive to 

this matter. This could maybe explain surprising similarity of 

AC loss of coils with rather different critical current [20]. Our 

calculations confirmed that for a coil with constant Jc the AC 

loss would be inversely proportional to Jc. 

Rather interesting is the result of numerical calculation 

showing that a turn from tape with reduced Jc could transfer 

the surplus dissipation to neighboring turns. At AC currents 

with amplitude approaching the critical current this would 

even make the actual dissipation in weaker turn lower 

compared to the case when all the turns are made from 

identical tape. Experimental verification of this theoretical 

prediction would be very desirable. 

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AC loss in pancake coil made from 12 mm wide REBCO tape

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